It is also the first time the club has reached this stage since the 2004/05 season, and was achieved thanks to two goals from Max Worsfold, another from Alfonso Tenconi and a host of saves from young goalkeeper Michael Edegbe, who was named man of the match.
There was an early chance for the visitors with Hutchinson allowed to come inside from the left and get in a shot, which Edegbe got down well to save at his near post.
But it took just nine minutes for the Rebels to open the scoring with Scott Davies taking a quick free kick which found Tenconi in space, and he thumped a composed finish past Lee Worgan.
It was almost two when Tenconi played the ball to Hunt who bore down on goal, but Worgan saved with his legs.
At the other end, Dominic Hutchinson was unlucky to see his low drive rebound back off the far post after he cut inside the penalty area from the left. Moments later, Edegbe had to make a good block at the feet of Charley Kendall.
But soon after, the Rebels doubled their lead. Worsfold stood over a free kick on the right flank, awarded after Jack Currie handled the ball, and with everyone expecting a cross, Worsfold whipped the ball over Worgan and into the far top corner.
The first half ended with Slough on top, and George Hunt almost added a third from the edge of the box, but Worgan got down well to save.
Eastbourne emerged from the half time break with a renewed purpose and Slough had Edegbe to thank as he saved from Kendall after the striker was played through one-on-one.
However, Slough were next to strike with Worsfold scoring his second. A long throw from Guy Hollis was cleared only as far as Freddie Grant, whose scuffed shot found its way into the six yard box where Worsfold pounced to convert the chance.
Within a minute though, Eastbourne found a route back into the game with Greg Luer firing across Edegbe and into the far corner.
Edegbe was called into action again on 77 minutes when he saved from Hutchinson at his near post.
But there was no way through for the visitors, and Slough will now look forward to Monday’s Fourth Round draw.
Image: Philip Benton
Slough Town Line Up
- 21 Michael Edegbe
- 3 Freddie Grant
- 4 Guy Hollis
- 5 Mark Nisbet
- 7 Max Worsfold
- 8 Matthew Lench 17
- 10 Alfonso Tenconi 19
- 12 George Hunt
- 14 Scott Davies
- 15 Aaron Kuhl
- 16 Sean Fraser
Eastbourne Borough Line up
Lee Worgan, Jake Elliott (Joel Rollinson 39), Jack Currie, Luke Nippard (Joshua Oyinsan 86), Steven James, James Hammond, Greg Luer, Charlie Walker (c), Dominic Hutchinson, Trent Mahorn, Charley Kendall. Unused substitutes: James Beresford, Charlie Towning, Charlie Lambert, Marcel Powell, Gregor Shaw.